On the homotopy type of the spaces of spherical knots in Rⁿ

2020/12/17 by Victor Turchin, Thomas Willwacher, Turchin, Victor +1
Mathematics · #18G85 #55P62 #58D10 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2012.09965

openalex publication_date 2020/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the spaces of embeddings S^m\hookrightarrow Rⁿ and those of long embeddings R^m\hookrightarrow Rⁿ, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. We find a natural fiber sequence relating these spaces. We also compare the L_∞-algebras of diagrams that encode their rational homotopy type, when the codimension n-m≥ 3.

Citations

• The rational homotopy of mapping spaces of E_n operads
• Real models for the framed little n-disks operads

Related

• On the rational homotopy type of embedding spaces of manifolds in Rⁿ
• Calculus of the embedding functor and spaces of knots
• A Weiss-Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots
• Haefliger's approach for spherical knots modulo immersions
• Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots